BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 90

⎧
⎨

2 U
V

1

V
2

V
2 U ≤

− t + for − t< 0 ,

2 U
V

1

x(t) ˙ = (3.138)

⎩ V

2

V
2 U .

t − for 0 ≤ t ≤

The time histories of the coordinate x(t) and the velocity x of the object

relative to the base are plotted in Figs. 3.7 and 3.8 in a normalized form. The
curve of Fig. 3.7 plots Ux/V 2 versus Ut/V and the curve of Fig. 3.8 plots
x/V versus Ut/V . The object starts moving relative to the base V/( 2 U)
time units before the anticipated shock and moves backward with the accel-
eration 0 when an instantaneous shock pulse is

applied to the base. At the instant just before the pulse, the coordinate of the
object attains the minimum value x

− U until the instant t =

V 2 /( 8 D) and the velocity is equal to
− V/ 2. Due to the pulse, the velocity jumps from

=

− V/ 2to V/ 2, after which

the object moves forward with the acceleration − U and at the time instant

Ux
V 2

0.05

0.00

- 0.05

- 0.10

- 0.13

- 0.5 0.0 0.5 1.0

Ut/V

FIGURE 3.7 Minimization of the rattlespace: Normalized time history of the relative
displacement of the object.

x
V

0.50

0.25

0.00

- 0.25

- 0.50

- 0.5 0.0 0.5 1.0

Ut/V

FIGURE 3.8 Minimization of the rattlespace: Normalized time history of the relative
velocity of the object.